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Double Angle Formula Calculator

Calculate sin(2θ), cos(2θ), and tan(2θ) using the double angle formulas.

Reviewed by Christopher FloiedPublished Updated

This free online double angle formula calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Results

sin(2θ)

0.866025

cos(2θ)

0.5

tan(2θ)

1.732051

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Double Angle Formula Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Double Angle Formula Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Double Angle Formula Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Double Angle Formula Calculator

The double angle formulas express trigonometric functions of 2θ in terms of functions of θ. These identities — sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos²(θ) − sin²(θ), and tan(2θ) = 2tan(θ)/(1 − tan²(θ)) — are fundamental in trigonometry and appear throughout mathematics, physics, and engineering. They are derived from the angle addition formulas by setting both angles equal to θ. Double angle formulas are used to simplify expressions, solve trigonometric equations, and evaluate integrals. In physics, they appear in the analysis of projectile motion (the range formula uses sin(2θ)), wave interference patterns, and rotational dynamics. This calculator computes all three double-angle values for any input angle.

The Math Behind It

The double angle formulas are special cases of the angle addition formulas. From sin(A + B) = sin A cos B + cos A sin B, setting A = B = θ gives sin(2θ) = 2sin(θ)cos(θ). From cos(A + B) = cos A cos B − sin A sin B, setting A = B = θ gives cos(2θ) = cos²(θ) − sin²(θ). The cosine double angle has three equivalent forms: cos(2θ) = cos²(θ) − sin²(θ) = 2cos²(θ) − 1 = 1 − 2sin²(θ). Rearranging the last two forms gives the power-reduction formulas: cos²(θ) = (1 + cos(2θ))/2 and sin²(θ) = (1 − cos(2θ))/2, which are essential for integration. The tangent double angle formula follows from tan(A + B) = (tan A + tan B)/(1 − tan A tan B). In projectile motion, the range R = v²sin(2θ)/g is maximized when 2θ = 90°, i.e., θ = 45°. Double angle formulas also appear in the rotation matrix and in Chebyshev polynomials: T₂(x) = 2x² − 1 mirrors cos(2θ) = 2cos²(θ) − 1.

Formula Reference

Double Angle – Sine

sin(2θ) = 2 sin(θ) cos(θ)

Variables: θ = angle

Double Angle – Cosine

cos(2θ) = cos²(θ) − sin²(θ)

Variables: θ = angle

Double Angle – Tangent

tan(2θ) = 2tan(θ) / (1 − tan²(θ))

Variables: θ = angle

Worked Examples

Example 1: Projectile range angle

Compute sin(2θ) for θ = 30° (used in range formula).

Step 1:sin(2 × 30°) = sin(60°)
Step 2:sin(60°) = √3/2
Step 3:Or using the formula: 2 sin(30°) cos(30°) = 2 × 0.5 × 0.866

sin(60°) ≈ 0.8660

Common Mistakes & Tips

  • !Confusing sin(2θ) with 2sin(θ) — they are not equal.
  • !Forgetting the three equivalent forms of cos(2θ).
  • !Not checking that tan(2θ) is undefined when tan²(θ) = 1 (i.e., θ = 45° or 225°).

Related Concepts

Half Angle Calculator

Formulas for sin(θ/2), cos(θ/2), and tan(θ/2).

Trig Functions

Calculate basic trig function values.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Half Angle Calculator

Frequently Asked Questions

What are the three forms of cos(2θ)?

cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. All three are equivalent.

When is the double angle formula for tangent undefined?

When tan²(θ) = 1, i.e., θ = 45° + n × 90° for any integer n.

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